(Thermal Dynamics and Statistical Mechanics) A molecule of hydrogen in its groun

Question

(Thermal Dynamics and Statistical Mechanics)

A molecule of hydrogen in its ground state can exist in two forms: ortho-hydrogen where the nuclear spins are parallel, resulting in a net spin; and para-hydrogen where the nuclear spins are antiparallel, resulting in no net spin. The ortho form of hydrogen has three spin states all of the same energy > 0, the para form has one state of zero energy. The hydrogen molecules form a solid made up of N molecules, with each molecule distinguishable from the rest by being localized on a lattice site. Assume that the spins of neighboring moelcules couple very weakly. The solid is thermally isolatted and has a fixed energy U = n where n is the number of molecules in the ortho state. Show that the entropy of the system in the microcanocnical ensemble is W = (N!/(n!(N-n)!)3n and obtain an expression for the temperature of the system. How does the number of molecules in the para state vary with temperature?

[The answer to the expression is n = 3/(3 + exp(/kBT)) but how do you get there?]

Explanation / Answer

given

> 0,

thermally isolatted and has a fixed energy U = n

the entropy of the system in the microcanocnical ensemble is W = (N!/(n!(N-n)!)3n

using

n = U - T S ------ 1

where S = kB logW

then

S = kB log ( (N!/(n!(N-n)!)3n )

substituting U and S values in equation ( 1 )

n = ( n - T kB log ( (N!/(n!(N-n)!)3n ) )

n = ( n - T kB log ( (N! ) - log ( (n!(N-n)!)3n ) ) -------2

here in the above equation log N! = N logN - N

and log n! = n logn - n

and also

log ( N - n )! = ( N - n ) log ( ( N - n ) ) - ( N - n )

substituting

log N! , log n! , log ( N - n )! values are in equation 2

n = ( n - T kB ( N logN - N ) - log ( n logn - n ) ( ( N - n ) log ( ( N - n ) ) - ( N - n ) ) 3n ) )

by solving the equation finally can be written as

n = 3 / ( 3 + exp ( / kBT ) )

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